1 Answer
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In both sections, you'll want to remove these parts specifically from the template:

<span class="icon"><a href="{Permalink}">{lang:Text}</a></span>

The above will remove the "text" phrase from text posts. The above will remove "photo" from photo posts.

<span class="icon"><a href="{Permalink}">{lang:Photo}</a></span>

Keep in mind that these are also the permalinks to the post themselves. That means if you wanted to show just that page you may have a harder time with the photo posts than the text posts are the permalink is still in the title.

If you want to add the permalink, you need this part:

<a href="{Permalink}">XYZ</a>

and replace the XYZ with the part of the post you want to create a link from. But you can just leave it off all the same.